where \(\langle j,k\rangle \)  denotes a summation over nearest-neighbor lattice sites and \(\sigma \in \lbrace 0,1\rbrace \)  labels the spin degree of freedom. The Fermi-Hubbard model represents a lattice system with nearest-neighbor interactions and, according to [1]}, can be simulated with \(\mathcal {O}\left(n^{1+1/p}\right)\)  gates using a \(p\) th-order Trotterization for a constant time and accuracy. On the other hand, recent work [2]} shows that the Trotterization algorithm has gate complexity \(\mathcal {O}\left(n\eta ^{1+1/p}\right)\)  when restricted to the \(\eta \) -electron manifold. By simultaneously using the sparsity of interactions, commutativity of the Hamiltonian and information about the initial state, we show in [sec:apphubbard]Section sec:apphubbard that \(\mathcal {O}\left(n\eta ^{1/p}\right)\)  gates suffices, improving both results for the Fermi-Hubbard model.Note however that this does not significantly improve the approach based on Lieb-Robinson bounds [3]}, since that approach has gate complexity \(\widetilde{\mathcal {O}}\left(n\right)\)  when using a high-precision quantum simulation algorithm as a subroutine.
